1,992 research outputs found
Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions
We describe the implementation of algorithms to construct and maintain
three-dimensional dynamic Delaunay triangulations with kinetic vertices using a
three-simplex data structure. The code is capable of constructing the geometric
dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points
is triangulated. Time evolution of the triangulation is not only governed by
kinetic vertices but also by a changing number of vertices. We use
three-dimensional simplex flip algorithms, a stochastic visibility walk
algorithm for point location and in addition, we propose a new simple method of
deleting vertices from an existing three-dimensional Delaunay triangulation
while maintaining the Delaunay property. The dual Dirichlet tessellation can be
used to solve differential equations on an irregular grid, to define partitions
in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly
nomenclature), referee suggestions included, typos corrected, bibliography
update
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
Essential Constraints of Edge-Constrained Proximity Graphs
Given a plane forest of points, we find the minimum
set of edges such that the edge-constrained minimum spanning
tree over the set of vertices and the set of constraints contains .
We present an -time algorithm that solves this problem. We
generalize this to other proximity graphs in the constraint setting, such as
the relative neighbourhood graph, Gabriel graph, -skeleton and Delaunay
triangulation. We present an algorithm that identifies the minimum set
of edges of a given plane graph such that for , where is the
constraint -skeleton over the set of vertices and the set of
constraints. The running time of our algorithm is , provided that the
constrained Delaunay triangulation of is given.Comment: 24 pages, 22 figures. A preliminary version of this paper appeared in
the Proceedings of 27th International Workshop, IWOCA 2016, Helsinki,
Finland. It was published by Springer in the Lecture Notes in Computer
Science (LNCS) serie
Improved Incremental Randomized Delaunay Triangulation
We propose a new data structure to compute the Delaunay triangulation of a
set of points in the plane. It combines good worst case complexity, fast
behavior on real data, and small memory occupation.
The location structure is organized into several levels. The lowest level
just consists of the triangulation, then each level contains the triangulation
of a small sample of the levels below. Point location is done by marching in a
triangulation to determine the nearest neighbor of the query at that level,
then the march restarts from that neighbor at the level below. Using a small
sample (3%) allows a small memory occupation; the march and the use of the
nearest neighbor to change levels quickly locate the query.Comment: 19 pages, 7 figures Proc. 14th Annu. ACM Sympos. Comput. Geom.,
106--115, 199
Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing
Motivated by the desire to cope with data imprecision, we study methods for
taking advantage of preliminary information about point sets in order to speed
up the computation of certain structures associated with them.
In particular, we study the following problem: given a set L of n lines in
the plane, we wish to preprocess L such that later, upon receiving a set P of n
points, each of which lies on a distinct line of L, we can construct the convex
hull of P efficiently. We show that in quadratic time and space it is possible
to construct a data structure on L that enables us to compute the convex hull
of any such point set P in O(n alpha(n) log* n) expected time. If we further
assume that the points are "oblivious" with respect to the data structure, the
running time improves to O(n alpha(n)). The analysis applies almost verbatim
when L is a set of line-segments, and yields similar asymptotic bounds. We
present several extensions, including a trade-off between space and query time
and an output-sensitive algorithm. We also study the "dual problem" where we
show how to efficiently compute the (<= k)-level of n lines in the plane, each
of which lies on a distinct point (given in advance).
We complement our results by Omega(n log n) lower bounds under the algebraic
computation tree model for several related problems, including sorting a set of
points (according to, say, their x-order), each of which lies on a given line
known in advance. Therefore, the convex hull problem under our setting is
easier than sorting, contrary to the "standard" convex hull and sorting
problems, in which the two problems require Theta(n log n) steps in the worst
case (under the algebraic computation tree model).Comment: 26 pages, 5 figures, 1 appendix; a preliminary version appeared at
SoCG 201
A Systematic Review of Algorithms with Linear-time Behaviour to Generate Delaunay and Voronoi Tessellations
Triangulations and tetrahedrizations are important geometrical discretization procedures applied to several areas, such as the reconstruction of surfaces and data visualization. Delaunay and Voronoi tessellations are discretization structures of domains with desirable geometrical properties. In this work, a systematic review of algorithms with linear-time behaviour to generate 2D/3D Delaunay and/or Voronoi tessellations is presented
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